existence of solutions – Optimization Online

Irreparable Linear Bilevel Problems with Rotations

Published: 2026/10/30

Bilevel Optimization bilevel optimization, Counterfactual explanations, existence of solutions, left-hand side parameterization, Set-valued continuity, Stability under perturbations

Classic linear bilevel optimization assumes that the upper-level decision impacts only the right-hand side of the lower-level constraints, which guarantees a solution under mild conditions. We study existence of solutions for linear bilevel problems with rotations (LBPR, for short), i.e., problems for which the upper-level decision modifies the left-hand side coefficients of the linear follower … Read more

Set Intersection Theorems and Existence of Optimal Solutions

Published: 2004/12/30

Constrained Nonlinear Optimization, Convex Optimization existence of solutions, frank-wolfe, quadratic programming, recession directions

The question of nonemptiness of the intersection of a nested sequence of closed sets is fundamental in a number of important optimization topics, including the existence of optimal solutions, the validity of the minimax inequality in zero sum games, and the absence of a duality gap in constrained optimization. We introduce the new notion of … Read more